Does a fully convolutional network share the same translation invariance properties we get from networks that use max-pooling?
All convolutional networks (with or without max-pooling) are translation-invariant (AKA spatially invariant) because their filters slide over every position in the image. This means that if a pattern that "matches" a filter is present anywhere in the image, then at least one neuron should activate.

Max-pooling, on the other hand, has nothing to do with spatial invariance. It's simply a regularization technique to help reduce the number of parameters later in the network by downsizing activation layers within the network. This can help combat overfitting, although it's not strictly necessary. Alternatively, neural networks can achieve the same effect by using a convolutional layer with a stride of 2 instead of 1.